1. 6 minutes
Warm-Up
For each parabola, find an equation for the axis of symmetry and the coordinates of the vertex. State whether the parabola opens up or down, and whether the y-coordinate of the vertex is the minimum or maximum value of the function. 1) 2)

2. 9.2 Parabolas Objectives:
Write and graph the standard equation of a parabola given sufficient information
Given an equation of a parabola, graph it and label the vertex, focus, and directrix

3. Parabolas
A parabola is defined in terms of a fixed point, called the focus,
A parabola is the set of all points P(x,y) in the plane whose distance to the focus
focus
and a fixed line,
called the directrix.
equals its distance to the directrix.
directrix
axis of symmetry

4. Horizontal Directrix
Standard Equation of a parabola with its vertex at the origin is
y = x2 1 4p x y
D(x, –p)
P(x, y)
F(0, p)
y = –p O
p > 0: opens upward
p < 0: opens downward
focus: (0, p)
directrix: y = –p
axis of symmetry: y-axis

5. Vertical Directrix
Standard Equation of a parabola with its vertex at the origin is
x = y2 1 4p x y
D(x, –p)
P(x, y)
F(p, 0)
x = –p O
p > 0: opens right
p < 0: opens left
focus: (p, 0)
directrix: x = –p
axis of symmetry: x-axis

6. Example 1 Graph . Label the vertex, focus, and directrix. Identify p.-4 -2 2 4
So, p = 1
Since p > 0, the parabola opens to the right.
Vertex: (0,0)
Focus: (1,0)
Directrix: x = -1

7. Example 1 Graph . Label the vertex, focus, and directrix. y x
Use a table to sketch a graph -4 -2 2 4 y x 2 4 -2 -4 1 4 1 4

8. Example 2
Write the standard equation of the parabola with its vertex at the origin and the directrix y = -6.
Since the directrix is below the vertex, the parabola opens up
Since y = -p and y = -6,
p = 6

9. Standard Equation of a Translated Parabola
Horizontal Directrix:
y – k = (x – h)2 1 4p
vertex: (h, k)
focus: (h, k + p)
directrix: y = k – p
axis of symmetry: x = h

10. Standard Equation of a Translated Parabola
Vertical Directrix:
x – h = (y - k)2 1 4p
vertex: (h, k)
focus: (h + p, k)
directrix: x = h - p
axis of symmetry: y = k

11. Example 3
Write the standard equation of the parabola with a focus at F(-3,2) and directrix y = 4.
The parabola opens downward, so the equation is of the form
y – k = (x – h)2 1 4p
vertex: (-3,3)
h = -3, k = 3
p = -1
y – 3 = (x + 3)2 1 -4

12. Practice
Write the standard equation of the parabola with its focus at F(-6,4) and directrix x = 2.

13. Example 4 Graph the parabola . Label the vertex, focus, and directrix.
-4( ) ( )-4
Isolate the y-terms
Complete the square

14. Example 4 Graph the parabola . Label the vertex, focus, and directrix.
x – h = (y - k)2 1 4p
directrix:
x = 0
vertex: (h, k)
= (1,-2)
Find p:
so, p = 1
focus:
F(2,-2)
focus:
= (2,-2)
directrix:
x = 0
vertex:
V(1,-2)

15. Example 4
Graph the parabola Label the vertex, focus, and directrix.
directrix:
x = 0 y x 2 -2 -4 -6
focus:
F(2,-2) 2 5 1 2 5
vertex:
V(1,-2)

16. Practice
Graph the parabola x2 – 6x + 6y + 18 = 0. Label the vertex, focus, and directrix.

17. Homework
worksheet